22.3.12 problem 6.47

Internal problem ID [4525]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 6. The Laplace Transform and Its Applications. Problems at page 291
Problem number : 6.47
Date solved : Tuesday, September 30, 2025 at 07:33:52 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=8 \left (t^{2}+t -1\right ) \operatorname {Heaviside}\left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.393 (sec). Leaf size: 52

ode:=diff(diff(y(t),t),t)+4*y(t) = 8*(t^2+t-1)*Heaviside(t-2); 
ic:=[y(0) = 1, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -9 \operatorname {Heaviside}\left (t -2\right ) \cos \left (-4+2 t \right )-5 \operatorname {Heaviside}\left (t -2\right ) \sin \left (-4+2 t \right )+\left (2 t^{2}+2 t -3\right ) \operatorname {Heaviside}\left (t -2\right )+\cos \left (2 t \right )+\sin \left (2 t \right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 54

ode=D[y[t],{t,2}]+4*y[t]==8*(t^2+t-1)*UnitStep[t-2]; 
ic={y[0]==1,Derivative[1][y][0] == 2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \cos (2 t)+\sin (2 t) & t\leq 2 \\ 2 t^2+2 t-9 \cos (4-2 t)+\cos (2 t)+5 \sin (4-2 t)+\sin (2 t)-3 & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 2.294 (sec). Leaf size: 116

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((-8*t**2 - 8*t + 8)*Heaviside(t - 2) + 4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 2 t^{2} \theta \left (t - 2\right ) + 2 t \left (1 - \cos {\left (2 t \right )}\right )^{2} \theta \left (t - 2\right ) - t \cos {\left (4 t \right )} \theta \left (t - 2\right ) - t \theta \left (t - 2\right ) + \left (4 t \theta \left (t - 2\right ) + 1\right ) \cos {\left (2 t \right )} + \sin {\left (2 t \right )} - 5 \sin {\left (2 t - 4 \right )} \theta \left (t - 2\right ) - 9 \cos {\left (2 t - 4 \right )} \theta \left (t - 2\right ) - 3 \theta \left (t - 2\right ) \]